J. Math. Biol. (2016) 72:255–281 DOI 10.1007/s00285-015-0884-1 Mathematical Biology Beyond clustering: mean-field dynamics on networks

© The Author(s) 2015. This article is published with open access at Springerlink.com Abstract Clustering is the propensity of nodes that share a common neighbour to be connected. It is ubiquitous in many networks but poses many modelling challenges. Clustering typically manifests itself by a higher than expected frequency of triangles, and this has led to the principle of constructing networks from such building blocks. This approach has been generalised to networks being constructed from a set of more exotic subgraphs. As long as these are fully connected, it is then possible to derive mean-field models that approximate epidemic dynamics well. However, there are virtually no results for non-fully connected subgraphs. In this paper, we provide a general and automated approach to deriving a set of ordinary differential equations, or mean-field model, that describes, to a high degree of accuracy, the expected values of system-level quantities, such as the prevalence of infection. Our approach offers a previously unattainable degree of control over the arrangement of subgraphs and network characteristics such as classical node degree, variance and clustering. The combination of these features makes it possible to generate families of networks with Electronic supplementary material The online version of this article (doi:10.1007/s00285-015-0884-1) contains supplementary material, which is available to authorized users